The award is likely to take Mr Dudley’s pay package for the year to about £3.7m if his basic salary, which is yet to be disclosed, remains at 2011 levels of $1.7m (£1.1m).

The share element of Mr Dudley’s bonus was disclosed in regulatory filings on Tuesday, exactly one week after BP revealed that its annual profits had halved in 2012, to $12bn (£7.6bn), weighed down by criminal penalties over the Gulf of Mexico disaster. Excluding one-off items, underlying replacement cost profit fell 19pc.

BP also disclosed on Tuesday that Mr Dudley could be eligible for further shares, worth up to £6.3m at current price, under the company’s long-term incentive plan (LTIP). He will receive a proportion of those shares in three years’ time, depending on the oil major’s performance between 2013 and 2015.

Mr Dudley's total pay package is likely to be lower this year because of the company’s poor performance relative to its peers between 2010 and 2012, which means potential share awards under the LTIP for the period are unlikely to have vested.

Mr Dudley's annual bonus for 2012 was judged against criteria including the company's safety record, its progress in "rebuilding trust" and "internal morale", as well as financial metrics such as operating cash flow and underlying replacement cost profit, according to its 2011 annual report.

BP executives must take one-third of their annual bonus in shares, and can also opt to take another third in shares. Under the so-called "deferred matching award" those share awards are then doubled by the company and will vest in three years' time so long as BP hits safety and environmental sustainability criteria.

BP's stock market filing showed Mr Dudley opted to take two-thirds of his annual bonus in shares. Including the matching shares, Mr Dudley was awarded American Depository Shares worth £2.1m based on the closing price on Monday, the day they were awarded. That implies a cash award for the remaining portion of his bonus of about £522,000.